[Paper Review] The effect of gas drag on the growth of protoplanets -- Analytical expressions for the accretion of small bodies in laminar disks

Chris W. Ormel, Hubert Klahr|arXiv (Cornell University)|Jul 6, 2010

Astrophysics and Star Formation StudiesPhysics and Astronomy55 references262 citations

TL;DR

This paper develops analytical expressions for the accretion of small particles by protoplanets in laminar protoplanetary disks, accounting for gas drag via a 2D circularly restricted three-body model with linear drag. It identifies three accretion modes—settling, hyperbolic, and three-body encounters—and shows that settling enables rapid growth of protoplanets above ~1,000 km by significantly increasing the impact radius, offering a fast growth channel in the outer disk.

ABSTRACT

Planetary bodies form by accretion of smaller bodies. It has been suggested that a very efficient way to grow protoplanets is by accreting particles of size <

Motivation & Objective

• To understand how gas drag influences the accretion of small particles (e.g., chondrules, boulders) by protoplanets in laminar disks.
• To identify and characterize distinct accretion modes—settling, hyperbolic, and three-body encounters—under gas drag.
• To derive analytical expressions for impact radii and accretion rates that match numerical simulations.
• To assess the viability of fragment sweepup as a fast-track growth mechanism for protoplanets in the outer solar system.

Proposed method

• Numerical integration of particle trajectories in a 2D circularly restricted three-body problem including linear gas drag force.
• Use of dimensionless parameters: headwind velocity (ζw) and Stokes number (St) to reduce the parameter space.
• Derivation of analytical recipes for impact radius in three distinct accretion regimes: settling, hyperbolic, and three-body encounters.
• Extension of results to 3D geometry to estimate accretion times for protoplanets via particle sweepup.
• Validation of analytical expressions against numerical simulations, with focus on accuracy across St and ζw values.
• Incorporation of gravitational focusing and Hill radius scaling (Rh = a(Mp/3M*)^{1/3}) to model protoplanet influence.

Experimental results

Research questions

• RQ1 How does gas drag alter the effective impact radius for small particles accreting onto protoplanets in a laminar disk?
• RQ2 What are the dominant accretion modes (settling, hyperbolic, three-body) under varying Stokes numbers and headwind velocities?
• RQ3 Can analytical expressions accurately predict impact radii across the full parameter space of St and ζw?
• RQ4 At what protoplanet size does the settling mechanism become significant and how does it accelerate growth?
• RQ5 How does radial drift of particles affect the viability of fragment sweepup as a growth mechanism?

Key findings

• The settling mechanism, where particles fall radially toward the protoplanet due to gas drag, leads to impact radii independent of protoplanet size, enabling efficient accretion.
• For protoplanets of ~1,000 km, the settling mode provides a much faster growth channel than traditional gravitational focusing, especially for St ~ 1 particles.
• Accretion of fragments on small protoplanets (≤50 km) is slow due to the particles being distributed over a thick layer, reducing collision probability.
• The three-body encounter mode, enhanced by gas drag, increases capture probability but contributes less significantly than settling for larger protoplanets.
• Analytical recipes for impact radius in all three modes show excellent agreement with numerical simulations, except possibly in the three-body regime.
• The model provides lower limits for accretion rates; atmospheric envelopes on protoplanets ≥0.1 M⊕ would further enhance accretion, but this is beyond the current scope.